To determine whether there is an outlier in the given data set (104, 117, 104, 136, 189, 109, 113, 104), we can follow these steps:
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Calculate the Mean: \[ \text{Mean} = \frac{(104 + 117 + 104 + 136 + 189 + 109 + 113 + 104)}{8} = \frac{976}{8} = 122 \]
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Calculate the Median:
- First, order the data set: 104, 104, 104, 104, 109, 113, 117, 136, 189.
- Since there are 8 data points, the median is the average of the 4th and 5th values: \[ \text{Median} = \frac{(104 + 109)}{2} = \frac{213}{2} = 106.5 \]
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Identify Outliers:
- We can use the IQR method to find outliers. First, we calculate Q1 (first quartile), Q3 (third quartile), and the IQR (interquartile range).
- Q1 is the median of the first half of the data (104, 104, 104, 104): Q1 = 104.
- Q3 is the median of the second half of the data (109, 113, 117, 136, 189): Q3 = 117.
- IQR = Q3 - Q1 = 117 - 104 = 13.
- Outlier boundaries:
- Lower Boundary = Q1 - 1.5 * IQR = 104 - 19.5 = 84.5
- Upper Boundary = Q3 + 1.5 * IQR = 117 + 19.5 = 136.5
Since 189 is greater than 136.5, it is considered an outlier.
- Best Measure of Central Tendency:
- The mean is affected by the outlier (189), making it less representative of the data set as a whole. The median, however, is not affected by outliers, making it a better measure of central tendency in this case.
Therefore, the final conclusion is:
Yes, outlier at 189. Median is the best measure.
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